Trisecting the Angle: Archimedes’ Method

Euclid’s insistence (c. 300 bc) on using only unmarked straightedge and compass for geometric constructions did not inhibit the imagination of his successors. Archimedes (c. 285–212/211 bc) made use of neusis (the sliding and maneuvering of a measured length, or marked straightedge) to solve one of the great problems of ancient geometry: constructing an angle that is one-third the size of a given angle.

Archimedes’ method of angle trisection.Encyclopædia Britannica, Inc.

Given ∠AOB, draw the circle with centre at O through the points A and B. Thus, OA and OB are radii of the circle and OA = OB.

Extend the ray AO indefinitely.

Now take a straightedge marked with the length of the circle’s radius and maneuver it (this is the neusis) into position to draw a line segment from B through a point C on the circle to a point D on the ray AO such that CD is equal to the circle’s radius; that is, CD = OC = OB = OA.